JEE Main 2024 (Online) 6th April Morning Shift

If \(A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1)\) and \(D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)\) are the vertices of a quadrilateral \(A B C D\), then its area is

The shortest distance between the lines \(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}\) and \(\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\) is

Let \(P\) be the point \((10,-2,-1)\) and \(Q\) be the foot of the perpendicular drawn from the point \(R(1,7,6)\) on the line passing through the points \((2,-5,11)\) and \((-6,7,-5)\). Then the length of the line segment \(P Q\) is equal ...

The interval in which the function \(f(x)=x^x, x>0\), is strictly increasing is

Let the area of the region enclosed by the curves \(y=3 x, 2 y=27-3 x\) and \(y=3 x-x \sqrt{x}\) be \(A\). Then \(10 A\) is equal to

If the second, third and fourth terms in the expansion of \((x+y)^n\) are 135, 30 and \(\frac{10}{3}\), respectively, then \(6\left(n^3+x^2+y\right)\) is equal to __________.

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are \(m\) and \(n\), respectively, then \(m+n^2\) is equal to

\(\int_\limits0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x \text { is equal to }\)

Let \(r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in \mathbb{N}\). Then the value of \(\sum_\limits{k=1}^{10} \frac{1}{7\left(r_k-1\right)}\) is equal to _________.

Let \(y=y(x)\) be the solution of the differential equation \(\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, x>0\) and \(y\left(e^{-1}\right)=0\). Then, \(y(e)\) is equal to

Let \(y=y(x)\) be the solution of the differential equation \(\left(1+x^2\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}\), \(y(1)=0\). Then \(y(0)\) is

$$\text { If } f(x)=\left\{\begin{array}{ll} x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0 & , x=0 \end{array}\right. \text {, then }$$

Let \(f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}\) be a differentiable function such that \(f^{\prime}(1)=\lim _\limits{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)\). Then $$\lim _\limits{a \rightarrow \infty} \frac{a(a+1)}{2}...

The function \(f(x)=\frac{x^2+2 x-15}{x^2-4 x+9}, x \in \mathbb{R}\) is

For \(n \in \mathrm{N}\), if \(\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^{-1} n=\frac{\pi}{4}\), then \(n\) is equal to ________.

For \(\alpha, \beta \in \mathbb{R}\) and a natural number \(n\), let $$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$$. Then \(2 A_{10}-A_8\) is

Let \(\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}\). If \(x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)\) for some \(x, y, z \in \mathbb{R}, x y z \neq 0\), then \(6 \alpha+4 \beta+\gamma\) is equal to _________.

Let \(C\) be the circle of minimum area touching the parabola \(y=6-x^2\) and the lines \(y=\sqrt{3}|x|\). Then, which one of the following points lies on the circle \(C\) ?

Let a conic \(C\) pass through the point \((4,-2)\) and \(P(x, y), x \geq 3\), be any point on \(C\). Let the slope of the line touching the conic \(C\) only at a single point \(P\) be half the slope of the line joining the points \(P\) and...

Let \(L_1, L_2\) be the lines passing through the point \(P(0,1)\) and touching the parabola \(9 x^2+12 x+18 y-14=0\). Let \(Q\) and \(R\) be the points on the lines \(L_1\) and \(L_2\) such that the \(\triangle P Q R\) is an isosceles tria...

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

A company has two plants \(A\) and \(B\) to manufacture motorcycles. \(60 \%\) motorcycles are manufactured at plant \(A\) and the remaining are manufactured at plant \(B .80 \%\) of the motorcycles manufactured at plant \(A\) are rated of ...

Let \(\alpha, \beta\) be the distinct roots of the equation \(x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}\) and \(a_n=\alpha^n+\beta^n\). Then the minimum value of \(\frac{a_{2023}+a_{2025}}{a_{2024}}\) is

Let \(x_1, x_2, x_3, x_4\) be the solution of the equation \(4 x^4+8 x^3-17 x^2-12 x+9=0\) and \(\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m\). Then the value of \(m\) is _________.

Let the first term of a series be \(T_1=6\) and its \(r^{\text {th }}\) term \(T_r=3 T_{r-1}+6^r, r=2,3\),
............ \(n\). If the sum of the first \(n\) terms of this series is $$\frac{1}{5}\left(n^2-12 n+39\right)\left(4 \cdot 6^n-5 \c...

Let the relations \(R_1\) and \(R_2\) on the set \(X=\{1,2,3, \ldots, 20\}\) be given by \(R_1=\{(x, y): 2 x-3 y=2\}\) and \(R_2=\{(x, y):-5 x+4 y=0\}\). If \(M\) and \(N\) be the minimum number of elements required to be added in \(R_1\) a...

Let \(A=\{n \in[100,700] \cap \mathrm{N}: n\) is neither a multiple of 3 nor a multiple of 4\(\}\). Then the number of elements in \(A\) is

The mean and standard deviation of 20 observations are found to be 10 and 2 , respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

Let a variable line of slope \(m>0\) passing through the point \((4,-9)\) intersect the coordinate axes at the points \(A\) and \(B\). The minimum value of the sum of the distances of \(A\) and \(B\) from the origin is

Let \(\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}\) and a vector \(\vec{c}\) be such that \(\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}\). If $$\vec{a} \cdot \vec{c}=1...